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Johann Heinrich Lambert (1728--1777) --- Historical Sketch

The ratio of the circumference of a circle to its diameter is the number called pi, the Greek letter $\pi $. This number has fascinated people down through the ages. In the middle grades, students learn that $\pi $ is approximately $3$, or better yet $22/7$, or better yet $355/113$, or still better $3.14159265$. All of these approximations are rational numbers, but $\pi $ itself is not a rational number: we cannot write $\pi =m/n$ with $m$ and $n$ integers.

Johann Heinrich Lambert (a French name, rhymes with eclair) is best known as the person who first proved that $\pi $ is not a rational number. This result was published in 1768. Since then, it has been shown that $\pi $ is not only irrational, it is transcendental: it is not the root of any polynomial with integer coefficients.

Lambert was born in France in 1728. His father was a tailor with seven children and there was not much money available for books or education. When Heinrich was seven, the family fled for religious regions to a free city protected by Switzerland. He attended school until he was 12, at which time he had to quit and assist his father. But he didn't quit studying; he pursued the study of scientific topics without instruction. He was soon sufficiently self-educated to be able to work gainfully as a tutor, and then later---at age 17---as assistant to the editor of a newspaper. In 1748, at age 20, he was hired to tutor a count's eleven-year-old grandson and his cousin. Lambert became known and was in due course elected to the Swiss Scientific Society in Basel. He worked for the Society and began to publish his scientific discoveries. His first article appeared in 1755. The next year, he was sent with the two boys, now nineteen, on a three-year tour of Europe as their tutor, where he met several scientists.

In 1758 and 1759 Lambert published three books on the properties of light. In 1760, on the recommendation of Euler, he was hired as Professor of Astronomy at the St. Petersburg Academy of Sciences, a position he held for two years. He continued his scientific and mathematical work in Berlin as a colleague of Euler and Lagrange, but he had differences with Euler and in 1766, he returned to St. Petersburg. He died ten years later, at age 49, having published some 150 scientific and mathematical works including results about non-Euclidean geometry.

Back now to what Lambert is most remembered for. In 1768, he proved that if $x$ is a nonzero rational number, then tan $x$ must be irrational. So, since $\tan \pi /4=1$, it follows that $\pi /4$ must be irrational, whence $\pi $ is irrational.

It took until 1882 to prove that $\pi $ is transcendental. That proof is due to Ferdinand von Lindemann (1852--1939).

If $\sin x$ is rational, must $x$ be rational? Must it be irrational?

If $\sin x$ is rational, must $\cos x$ be rational? Must it be irrational? Can both be rational? Irrational?


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Copyright © 2006 Darel Hardy, Fred Richman, Carol Walker, Robert Wisner. All rights reserved. Except upon the express prior permission in writing, from the authors, no part of this work may be reproduced, transcribed, stored electronically, or transmitted in any form by any method.

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